What is uncertainty in measuring the volume of a cylinder? By the equation $V=\pi r^2 h$ it would suffice for one to measure both the radius and the height to measure the uncertainty in  $V$. Now suppose the absolute uncertainty in measuring the radius is $\delta r$ and the absolute uncertain in measuring the height is $\delta h$ , then what will be the absolute uncertainty of volume $\delta V$. I know that the uncertainty of $r^2$ is $2\delta r$ and when uncertainties are multiplied then the relative uncertainties add up. Therefore, I reckon that the following equation should hold: 
$${\delta V \over V}={2\delta r\over r^2}+{\delta h\over h}$$
Is this correct?  
 A: Assume first you only have an error $\delta r$ in r and know h precisely. Then it is correct to say:
$${\delta V \over V}={2\delta r\over r}$$
This follows from the fraction error rule.
Similarly if r is known precisely and we have an error $\delta h$ in h then:
$${\delta V \over V}={\delta h\over h}$$
So the question is really how to combine these errors. Now the fundamental assumption that is probably valid here is to assume independence of the errors. Intuitively this means that you measure the two separately and one error wouldn't influence the other.
Under this assumption we can write (see e.g. wikipedia):
$$ {\delta V \over V}=\sqrt{\left({2\delta r\over r}\right)^2 + \left({\delta h\over h}\right)^2}$$
i.e. independent errors are added in quadrature.
These rules often seem mysterious to undergraduate students and they really are badly explained in most courses. If you want to know more on the foundations of error theory can recommend this gem of a book by D. Sivia or Jayne's textbook on probability theory.
A: What you can notice about the equation that you have written is that the units are not correct in every term.  For parameters that are independent, you can do standard error propagation in the following way.  If you have a function $f$ which is dependent on multiple parameters $x_i$, then the error in $f$ is given by $$\sigma_f = \sqrt{\sum_i\left(\frac{df}{dx_i}\right)^2\sigma_{x_i}^2}$$
In your case, with $V=\pi r^2h$, you can perform this differentiation and obtain the error in $V$ as follows:
$$\sigma_V = \sqrt{4\pi^2 r^2h^2\sigma_r^2+\pi^2 r^4\sigma_h^2}$$
